reserve x,y,z,X,Y for set;
reserve X,Y for non empty set,
  f for Function of X,Y;
reserve X, Y for non empty set,
  F for (BinOp of Y),
  B for (Element of Fin X),
  f for Function of X,Y;

theorem Th29:
  F is idempotent & F is commutative & F is associative & F is
having_a_unity implies for x being Element of X holds F $$(B \/ {.x.}, f) = F.(
  F $$(B,f),f.x)
proof
  assume that
A1: F is idempotent and
A2: F is commutative & F is associative;
  assume
A3: F is having_a_unity;
  let x be Element of X;
A4: {} = {}.X;
  now
    assume
A5: B = {};
    hence F $$(B \/ {.x.}, f) = f.x by A2,Th14
      .= F.(the_unity_wrt F, f.x) by A3,Th12
      .= F.(F $$(B,f),f.x) by A2,A3,A4,A5,Th28;
  end;
  hence thesis by A1,A2,Th17;
end;
